Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C22×A4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | A4×M16 |
| unambiguous semidirect product | (C22×M16)⋊C3 |
| Fitting, Sylow | C22×M16 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×A4 |
| commutator, socle | C23 | C2×C3×C4 |
| Fitting | C22×M16 | C3 |
| prime radical | C22×A4 | C4 |
| cosocle | C22×C4 | C22×C3 |
| hypercenter | M16 | A4 |
| hypocenter | C22 | C3×M16 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C4 ◁ M16 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,3,4)(6,9)(10,12);(1,4)(2,3)(5,10,11,6,8,12,7,9)40 conjugacy classes — view the full character table.
| maximal subgroup of | #73114, #73115, #73116, … |
| maximal normal subgroup of | #73114, #73115, #73116, … |
| maximal quotient of | A4×#97, A4×C22⋊C8, A4×C4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1011);